STUDY OF THE HEATING DYNAMICS OF STEEL PRODUCTS

V. V. Salomatov, A. A. Torlopov

(Submitted by the Department of Boiler Engineering and Boiler Installations)

The compilation of optimal regime charts for heating steel billets and ingots is possible only when accurate information is available on the kinetics of the processes occurring in this case. As a rule, engineering practice does not always possess a sufficiently complete volume of reliable data on physical and technological parameters; therefore it is necessary to confine oneself to certain discrete information, on the basis of whose analysis a conclusion is usually drawn about the quality of heating.

The study of the thermophysics of the interconnected system—metallurgical furnace–metal—is a very complex problem, which, when studied, breaks down into two independent tasks: the external one, in which the formation of heat-exchange conditions is analyzed, and the internal one, in which the laws of heat transfer within the heated product under a prescribed heat flux at the boundary surface are considered. Deliberately restricting consideration to only the internal problem, as the most important from the standpoint of obtaining high-quality metal, heating practice defines as reference indicators the heating time of the billet, the magnitude of the maximum temperature difference over the ingot cross section, and the moment at which it occurs.

Overstating the norms for heating time reduces the possible productivity of furnaces and increases fuel consumption. Ignorance of the magnitudes of the maximum temperature difference over the billet cross section and of the fixed moment of its occurrence leads to an incorrect assignment of the heating regime, dangerous in individual cases from the standpoint of the temperature stresses that arise.

These indicators can become decisive only with such methods for calculating the temperature field as take into account with sufficient accuracy not only the heating conditions (the temperature of the external medium, the presence of radiant and convective heat flux), but also the phenomena occurring in the heated body itself (changes in thermophysical characteristics, the presence of structural transformations).

The present work is carried out from the standpoint of a theoretical study of the heating dynamics of steel products, with the aim of obtaining a sufficiently simple and reliable method for calculating the indicated indicators.

It is known that at present there still do not exist analytical relationships for calculating the nonstationary distribution of temperatures in bodies with variable thermophysical characteristics and phase-

…with transformations, subjected to the simultaneous action of radiation and convection. The difficulty here is that the heat-conduction problem being analyzed belongs to the class of doubly nonlinear problems. Therefore, a problem of this kind can be solved only with the aid of numerical methods.

We solved such a problem on the “Minsk-1” electronic computer [1]. Extensive numerical material was obtained on the temperature fields during the heating of products made of carbon and austenitic steels in the form of a plate and a cylinder, with the process parameters specified over wide ranges.

Fig. 1. Dependence of the heating time of products made of carbon steels on the criteria $\Theta_0$, $\mathrm{Sk}$, and $\mathrm{Bi}/\mathrm{Sk}$ for an infinite plate ($\Delta\Theta = 0.012$).

Fig. 2. Dependence of the heating time of products made of austenitic-type steels on the criteria $\Theta_0$, $\mathrm{Sk}$, and $\mathrm{Bi}/\mathrm{Sk}$ for an infinite plate ($\Delta\Theta = 0.012$).

Fig. 3. Dependence of the heating time of products made of carbon steels on the criteria $\Theta_0$, $\mathrm{Sk}$, and $\mathrm{Bi}/\mathrm{Sk}$ for an infinite cylinder ($\Delta\Theta = 0.012$).

Fig. 4. Dependence of the heating time of products made of austenitic-type steels on the criteria $\Theta_0$, $\mathrm{Sk}$, and $\mathrm{Bi}/\mathrm{Sk}$ for an infinite cylinder ($\Delta\Theta = 0.012$).

An objective criterion for the end of heating is the temperature difference over the cross section of the product. For each operation in the processing of billets and ingots there is a definite maximum permissible degree of nonuniformity of heating. Knowing the quantitative relation between the end-of-heating time and the criteria $\Theta_0$, $\mathrm{Sk}$, $\mathrm{Bi}/\mathrm{Sk}$ for a specified degree of temperature nonuniformity over the metal cross section, it is possible to construct calculation diagrams. Figs. 1, 2, 3, 4 present such depende…

… for calculating the heating time of bodies that can, in practice, be taken as infinite plates and infinite cylinders.

As can be seen from the graphs presented, the value of the criterion \(\mathrm{Fo}\) corresponding to the end of heating undergoes substantial changes up to the value \(\mathrm{Sk}=1\) for products made of carbon steel and the value \(\mathrm{Sk}=2\) for austenitic steel. Beyond these values of the criterion \(\mathrm{Sk}\), only a slight change is observed in the analyzed value \(\mathrm{Fo}\) as a function of the external heat-transfer criteria.

The convective component of the heat flux accelerates the heating of products with varying thermophysical characteristics the more intensively, the smaller the value of the radiation criterion \(\mathrm{Sk}\). In calculations, however, it makes sense to take it into account when the values of the criterion \(\mathrm{Sk}\) do not exceed 1 or 2, respectively, for ingots and billets made of carbon and austenitic steel.

Fig. 5. Change in the maximum temperature difference over the cross section of a product made of carbon steels as a function of the criteria \(\Theta_0\), \(\mathrm{Sk}\), and \(\mathrm{Bi}/\mathrm{Sk}\) for an infinite plate and cylinder.

As for the influence of \(\Theta_0\) at the end of heating, one may note an almost linear decrease in the value of the criterion \(\mathrm{Fo}\) at the end of heating with increasing \(\Theta_0\), for the same values of \(\mathrm{Sk}\) and \(\mathrm{Bi}/\mathrm{Sk}\).

Fig. 6. Change in the maximum temperature difference over the cross section of a product made of austenitic-type steels as a function of the criteria \(\Theta_0\), \(\mathrm{Sk}\), and \(\mathrm{Bi}/\mathrm{Sk}\) for an infinite plate and cylinder.

For prescribing a safe heating regime, as indicated above, the determining role is played by the magnitude of the maximum temperature difference over the cross section of the metal. In view of the difficulties of directly measuring this parameter under production conditions, it becomes necessary to have in advance an idea of the value \(\Delta\Theta_m\) and of the moment

its attainment under various heating regimes. Having numerical data on the temperature fields in the heated products, one can establish quantitative dependences of \(\Delta \Theta_m\) on the criteria \(\Theta_0\), \(Sk\), and \(Bi/Sk\). Such dependences are presented in Figs. 5 and 6.

On the basis of an analysis of the results obtained, it can be stated that, in the heating of billets and ingots with varying thermal coefficients, the magnitude of the maximum temperature difference, under otherwise identical conditions, is invariant with respect to the shape of the product. Characteristically, the same feature is observed when calculating the heating of products with constant thermophysical characteristics [2]. Therefore Figs. 5 and 6 can be used both for an unlimited plate and for an infinite cylinder.

For products made of austenitic-type steels, a linear dependence of \(\Delta \Theta_m\) on \(\Theta_0\) is observed; for products made of carbon steel this regularity occurs at values of the criterion \(Sk\) greater than 4.

The values of \(\Delta \Theta_m\), for identical criteria \(\Theta_0\) and \(Sk\), are in a linear relationship with the ratio \(Bi/Sk\).

Fig. 7. Dependence of the instant of occurrence of \(\Delta \Theta_m\) on the external heat-transfer criteria for an unlimited plate made of carbon steels.

Fig. 8. Dependence of the instant of occurrence of \(\Delta \Theta_m\) on the external heat-transfer criteria for an unlimited plate made of austenitic-type steels.

Fig. 9. Dependence of the instant of occurrence of \(\Delta \Theta_m\) on the external heat-transfer criteria for an infinite cylinder made of carbon steels.

Fig. 10. Dependence of the instant of occurrence of \(\Delta \Theta_m\) on the external heat-transfer criteria for an infinite cylinder made of austenitic-type steels.

Knowing the magnitudes of the external heat-transfer criteria, the instant of occurrence of the maximum temperature difference over the cross section of ingots is determined from Figs. 7, 8, 9, 10. The investigation shows that the instant at which \(\Delta \Theta_m\) is reached is not the same for bodies of different shape and material. Thus, when heating a plate made of austenitic-type steel, it occurs at values

$\mathrm{Fo}=0.07 \div 0.3$; and for a cylinder—at $\mathrm{Fo}=0.05 \div 0.2$ within the range of variation of the criterion $\mathrm{Sk}=0.5—4$. When products made of carbon steel are heated, the time at which $\Delta\Theta_m$ occurs lies within the ranges $\mathrm{Fo}=0.1 \div 0.6$ and $\mathrm{Fo}=0.1 \div 0.55$, respectively, for the plate and the cylinder, when the criterion $\mathrm{Sk}=0.25 \div 2$.

For materials made of carbon steel, at values of the criterion $\mathrm{Sk}<0.75$, in principle $\Delta\Theta_m$ is found in the zone of plastic deformations; however, since the greatest danger to the continuity of the metal is presented by the initial stage of heating, when the material still possesses elastic properties, the determining quantity must be taken to be the maximum temperature difference over the cross section that arises precisely in this zone. Figs. 7 and 9 were constructed taking this circumstance into account.

Conclusions

1. For the first time, on the basis of extensive numerical material on temperature fields, an analysis is given of the heating dynamics of steel products with variable thermophysical characteristics and structural transformations, heated jointly by radiation and convection.

2. An engineering method is proposed for determining indicators important for practice: the heating time, the maximum temperature difference over the cross section of steel products, and the moment at which it is reached.

PRINCIPAL NOTATION

$\mathrm{Fo}_*$ — Fourier criterion corresponding to the moment at which the maximum temperature difference over the cross section of the product occurs;

$\Delta\Theta_m$ — maximum temperature difference over the cross section of the product;

$\Delta\Theta$ — degree of nonuniformity of heating of bodies, equal to the difference between the temperatures of the surface and the center of the product at the end of heating.

The remaining notation is standard.

REFERENCES

1. V. V. Salomatov, A. A. Torlopov. Heating of metal products by radiative-convective heat with allowance for the variability of thermal coefficients and the thermal effect of structural transformations, article in the present collection.

2. G. P. Ivantsov. Heating of Metal. Metallurgizdat, 1948.